Positive expansiveness versus network dimension in symbolic dynamical systems
Abstract
A `symbolic dynamical system' is a continuous transformation F:X-->X of a closed perfect subset X of AV, where A is a finite set and V is countable. (Examples include subshifts, odometers, cellular automata, and automaton networks.) The function F induces a directed graph structure on V, whose geometry reveals information about the dynamical system (X,F). The `dimension' dim(V) is an exponent describing the growth rate of balls in the digraph as a function of their radius. We show: if X has positive entropy and dim(V)>1, and the system (AV,X,F) satisfies minimal symmetry and mixing conditions, then (X,F) cannot be positively expansive; this generalizes a well-known result of Shereshevsky about multidimensional cellular automata. We also construct a counterexample to a version of this result without the symmetry condition. Finally, we show that network dimension is invariant under topological conjugacies which are Holder-continuous.